Exponential stability in linear viscoelasticity
Author:
Vittorino Pata
Journal:
Quart. Appl. Math. 64 (2006), 499-513
MSC (2000):
Primary 35B40, 45K05, 45M10, 47D06
DOI:
https://doi.org/10.1090/S0033-569X-06-01010-4
Published electronically:
April 6, 2006
MathSciNet review:
2259051
Full-text PDF Free Access
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Abstract: We address the study of the asymptotic behavior of solutions to an abstract integrodifferential equation modeling linear viscoelasticity. Framing the equation in the past history setting, we analyze the exponential stability of the related semigroup $S(t)$ with dependence on the convolution kernel, providing a more general sufficient condition than the usual one present in the literature.
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CP V.V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal. 46 (2006), 251–273.
- Monica Conti and Vittorino Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal. 4 (2005), no. 4, 705–720. MR 2172716, DOI https://doi.org/10.3934/cpaa.2005.4.705
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AREY J.A.D. Appleby and D.W. Reynolds, On necessary and sufficient conditions for exponential stability in linear Volterra integro-differential equations, J. Integral Equations Appl. 16 (2004), 221–240.
CP V.V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal. 46 (2006), 251–273.
CONPAT M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal. 4 (2005), 705–720.
DAF1 C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297–308.
DAF2 C.M. Dafermos, Contraction semigroups and trend to equilibrium in continuum mechanics, in “Applications of Methods of Functional Analysis to Problems in Mechanics” (P. Germain and B. Nayroles, Eds.), pp. 295–306, Lecture Notes in Mathematics, no. 503, Springer-Verlag, Berlin-New York, 1976.
DAT R. Datko, Extending a theorem of A.M. Liapunov to Hilbert space, J. Math. Anal. Appl. 32 (1970), 610–616.
FL M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal. 116 (1991), 139–152.
FM M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, no. 12, SIAM, Philadelphia, 1992.
FP M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal. 81 (2002), 1245–1264.
GL C. Giorgi and B. Lazzari, On the stability for linear viscoelastic solids, Quart. Appl. Math. 55 (1997), 659–675.
GPR C. Giorgi, J.E. Muñoz Rivera, and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl. 260 (2001), 83–99.
Terreni M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in “Evolution Equations, Semigroups and Functional Analysis” (A. Lorenzi and B. Ruf, Eds.), pp. 155–178, Progr. Nonlinear Differential Equations Appl., no. 50, Birkhäuser, Boston, 2002.
LZ0 Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math. 54 (1996), 21–31.
LZ Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes in Mathematics, no. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999.
RIV J.E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math. 52 (1994), 629–648.
PZ V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. 11 (2001), 505–529.
PAZ A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
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Additional Information
Vittorino Pata
Affiliation:
Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
MR Author ID:
358540
Email:
pata@mate.polimi.it
Keywords:
Linear viscoelasticity,
memory kernels,
contraction semigroups,
exponential stability
Received by editor(s):
November 14, 2005
Published electronically:
April 6, 2006
Article copyright:
© Copyright 2006
Brown University
The copyright for this article reverts to public domain 28 years after publication.
